Maxwell's equations facts for kids
Maxwell's equations describe how electric charges and electric currents create electric and magnetic fields. They describe how an electric field can generate a magnetic field, and vice versa.
In the 1860s James Clerk Maxwell published equations that describe how charged particles give rise to electric and magnetic force per unit charge. The force per unit charge is called a field. The particles could be stationary or moving. These, and the Lorentz force equation, give everything one needs to calculate the motion of classical particles in electric and magnetic fields.
The first equation allows one to calculate the electric field created by a charge. The second allows one to calculate the magnetic field. The other two describe how fields 'circulate' around their sources. Magnetic fields 'circulate' around electric currents and time varying electric fields: Ampère's law with Maxwell's extension, while electric fields 'circulate' around time varying magnetic fields: Faraday's law.
Maxwell's Equations in the classical forms
Name  Differential form  Integral form 

Gauss' law:  
Gauss' law for magnetism (absence of magnetic monopoles): 

Faraday's law of induction:  
Ampère's law (with Maxwell's extension): 
where the following table provides the meaning of each symbol and the SI unit of measure:
Symbol  Meaning  SI Unit of Measure 

electric field  volt per metre  
magnetic field strength  ampere per metre  
electric displacement field  coulomb per square metre  
magnetic flux density also called the magnetic induction. 
tesla, or equivalently, weber per square metre 

free electric charge density, not counting the dipole charges bound in a material. 
coulomb per cubic metre  
free current density, not counting polarization or magnetization currents bound in a material. 
ampere per square metre  
differential vector element of surface area A, with very small magnitude and direction normal to surface S 
square meters  
differential element of volume V enclosed by surface S  cubic meters  
differential vector element of path length tangential to contour C enclosing surface c  meters  
instantaneous velocity of the line element defined above (for moving circuits).  meters per second 
and
 is the divergence operator (SI unit: 1 per metre),
 is the curl operator (SI unit: 1 per metre).
Covariant Formulation
There are only two covariant Maxwell Equations, because the covariant field vector includes the electrical and the magnetical field.
Mathematical note: In this section the abstract index notation will be used.
In special relativity, Maxwell's equations for the vacuum are written in terms of fourvectors and tensors in the "manifestly covariant" form. This has been done to show more clearly the fact that Maxwell's equations (in vacuum) take the same form in any inertial coordinate system. This is the "manifestly covariant" form:
 ,
and
The second equation is the same as:
Here is the 4current, is the field strength tensor (written as a 4 × 4 matrix), is the LeviCivita symbol, and is the 4gradient (so that is the d'Alembertian operator). (The in the first equation is implicitly summed over, according to Einstein notation.) The first tensor equation says the same thing as the two inhomogeneous Maxwell's equations: Gauss' law and Ampere's law with Maxwell's correction. The second equation say the same thing as the other two equations, the homogeneous equations: Faraday's law of induction and the absence of magnetic monopoles.
can also be described more explicitly by this equation: (as a contravariant vector), where you get from the charge density ρ and the current density . The 4current is a solution to the continuity equation:
In terms of the 4potential (as a contravariant vector) , where φ is the electric potential and is the magnetic vector potential in the Lorentz gauge , F can be written as:
which leads to the 4 × 4 matrix rank2 tensor:
The fact that both electric and magnetic fields are combined into a single tensor shows the fact that, according to relativity, both of these are different parts of the same thing—by changing frames of reference, what looks like an electric field in one frame can look like a magnetic field in another frame, and the other way around.
Using the tensor form of Maxwell's equations, the first equation implies
(See Electromagnetic fourpotential for the relationship between the d'Alembertian of the fourpotential and the fourcurrent, expressed in terms of the older vector operator notation).
Different authors sometimes use different sign conventions for these tensors and 4vectors (but this does not change what they mean).
and are not the same: they are related by the Minkowski metric tensor : . This changes the sign of some of F's components; more complex metric dualities can be seen in general relativity.
Images for kids

In a geomagnetic storm, a surge in the flux of charged particles temporarily alters Earth's magnetic field, which induces electric fields in Earth's atmosphere, thus causing surges in electrical power grids. (Not to scale.)

Magneticcore memory (1954) is an application of Ampère's law. Each core stores one bit of data.